It is shown that time asymmetry is essential for deriving thermodynamic law and arises from the turnover of energy while reducing its information content and driving entropy increase. A dynamically interpreted principle of least action enables time asymmetry and time flow as a generation of action and redefines useful energy as an information system which implements a form of acting information. This is demonstrated using a basic formula, originally applied for time symmetry/energy conservation considerations, relating time asymmetry (which is conventionally denied but here expressly allowed), to energy behaviour. The results derived then explained that a dynamic energy is driving time asymmetry. It is doing it by decreasing the information content of useful energy, thus generating action and entropy increase, explaining action-time as an information phenomenon. Thermodynamic laws follow directly. The formalism derived readily explains what energy is, why it is conserved (1st law of thermodynamics), why entropy increases (2nd law) and that maximum entropy production within the restraints of the system controls self-organized processes of non-linear irreversible thermodynamics. The general significance of the principle of least action arises from its role of controlling the action generating oriented time of nature. These results contrast with present understanding of time neutrality and clock-time, which are here considered a source of paradoxes, intellectual contradictions and dead-end roads in models explaining nature and the universe.
In this article, a finite volume element algorithm is presented and discussed for the numerical solutions of a time-fractional nonlinear fourth-order diffusion equation with time delay. By choosing the second-order spatial derivative of the original unknown as an additional variable, the fourth-order problem is transformed into a second-order system. Then the fully discrete finite volume element scheme is formulated by using L1approximation for temporal Caputo derivative and finite volume element method in spatial direction. The unique solvability and stable result of the proposed scheme are proved. A priori estimate of L2-norm with optimal order of convergence O(h2+τ2−α)where τand hare time step length and space mesh parameter, respectively, is obtained. The efficiency of the scheme is supported by some numerical experiments.
This paper presents space-time continuous and time discontinuous Galerkin schemes for solving nonlinear time-fractional partial differential equations based on B-splines in time and non-uniform rational B-splines (NURBS) in space within the framework of Iso-geometric Analysis. The first approach uses the space-time continuous Petrov-Galerkin technique for a class of nonlinear time-fractional Sobolev-type equations and the optimal error estimates are obtained through a concise equivalence analysis. The second approach employs a generalizable time discontinuous Galerkin scheme for the time-fractional Allen-Cahn equation. It first transforms the equation into a time integral equation and then uses the discontinuous Galerkin method in time and the NURBS discretization in space. The optimal error estimates are provided for the approach. The convergence analysis under time graded meshes is also carried out, taking into account the initial singularity of the solution for two models. Finally, numerical examples are presented to demonstrate the effectiveness of the proposed methods.
This paper aims to define the concept of time and justify its properties within the universal context, shedding new light on the nature of time. By employing the concept of the extrinsic universe, the paper explains the observable universe as the three-dimensional surface of a four-dimensional 3-sphere (hypersphere), expanding at the speed of light. This expansion process gives rise to what we perceive as time and its associated aspects, providing a novel interpretation of time as a geometric property emerging from the dynamics of the universe’s expansion. The work offers insights into how this extrinsic perspective can address phenomena such as the universe’s accelerated expansion and dark matter, aligning the model with current observational data.
Fractional-order time-delay differential equations can describe many complex physical phenomena with memory or delay effects, which are widely used in the fields of cell biology, control systems, signal processing, etc. Therefore, it is of great significance to study fractional-order time-delay differential equations. In this paper, we discuss a finite volume element method for a class of fractional-order neutral time-delay differential equations. By introducing an intermediate variable, the fourth-order problem is transformed into a system of equations consisting of two second-order partial differential equations. The L1 formula is used to approximate the time fractional order derivative terms, and the finite volume element method is used in space. A fully discrete format of the equations is established, and we prove the existence, uniqueness, convergence and stability of the solution. Finally, the validity of the format is verified by numerical examples.
